Since \(G\)  is a semidirect product of \(B\)  and \(L\) , it suffices to show that the hypotheses in Proposition REF  hold.
Since \(\mathfrak {h}\)  is a Cartan subalgebra, \(\mathfrak {h}+\mathfrak {b}_\infty = \mathfrak {b}\) , where \(\mathfrak {b}_\infty \)  is the intersection of the lower central series of \(\mathfrak {b}\)  [1]}. Since \(\mathfrak {b}\)  is solvable, it is contained in
the solvable radical \(\mathfrak {r}\)  of \(\mathfrak {g}\) . Therefore \(\mathfrak {b}_\infty \)  is contained in \(\mathfrak {r}_\infty \) , the intersection of the lower central series of \(\mathfrak {r}\) . Further, \(\mathfrak {r}_\infty \subset \mathfrak {e}\) ; see (REF ). Since in turn \(\mathfrak {e}\subset \mathfrak {n}^{\prime }\) , we have \(\mathfrak {n}^{\prime }+\mathfrak {h} = \mathfrak {b}\) .
Finally, \(\mathfrak {h}\) , \(\mathfrak {n}^{\prime }\)  and \(\mathfrak {b}/\mathfrak {n}^{\prime }\)  are nilpotent: \(\mathfrak {h}\)  being a Cartan subalgebra, \(\mathfrak {n}^{\prime }\)  by the hypothesis, and \(\mathfrak {b}/\mathfrak {n}^{\prime }\)  because so is \(\mathfrak {b}/\mathfrak {e}\)  [2]}. Thus all the hypotheses in Proposition REF  hold and this completes the proof.
